7. Prove or disprove each of the following. State clearly whether it is true or false. (a) (4 points) Given any two straight lines 41 and 62 in R3, there exist a pair of parallel planes II, and II2 (possibly the same plane) such that II1 contains 1, and I12 contains &2. (b) (4 points) Let f, g : R2 - R be two functions. If lim f (x) = lim g(x) (with x-a x -+a both exist) for any a E R2, then f and g are the same function. (c) (4 points) Let f : R2 - R be a continuous function. Define g(x) = f(x, 0) and h(y) = f(0, y). If g has a local minimum at x = 0 and h has a local minimum at y = 0, then (0, 0) is a point of local minimum of f. (d) (4 points) The vector field F : R3 - R3 defined by F(x, y, z) = (y3, 23, 23) is a gradient field